Geometry and Computing Series Editors Herbert Edelsbrunner doc

Geometry and Computing Series Editors Herbert Edelsbrunner Leif Kobbelt Konrad Polthier Editorial Advisory Board Jean-Daniel Boissonnat Gunnar Carlsson Bernard Chazelle Xiao-Shan Gao Craig Gotsman Leo Guibas Myung-Soo Kim Takao Nishizeki Helmut Pottmann Roberto Scopigno Hans-Peter Seidel Steve Smale Peter Schră der o Dietrich Stoyan Jean-Marie Morvan Generalized Curvatures With 107 Figures 123 Jean-Marie Morvan ´ Universite Claude Bernard Lyon Institut Camille Jordan ˆ Batiment Jean Braconnier 43 bd du 11 Novembre 1918 69622 Villeurbanne Cedex France morvanjeanmarie@yahoo.fr On the cover, the data of Michelangelo's head are courtesy of Digital Michelangelo Project, the image of Michelangelo's head with the lines of curvatures are courtesy of the GEOMETRICA project-team from INRIA ISBN 978-3-540-73791-9 e-ISBN 978-3-540-73792-6 Springer Series in Geometry and Computing Library of Congress Control Number: 2008923176 Mathematics Subjects Classification (2000): 52A, 52B, 52C, 53A, 53B, 53C, 49Q15, 28A33, 28A75, 68R c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: deblik, Berlin Printed on acid-free paper springer.com Contents Introduction 1.1 Two Fundamental Properties 1.2 Different Possible Classifications 1.3 Part I: Motivation 1.4 Part II: Background – Metric and Measures 1.5 Part III: Background – Polyhedra and Convex Subsets 1.6 Part IV: Background – Classical Tools on Differential Geometry 1.7 Part V: On Volume 1.8 Part VI: The Steiner Formula 1.9 Part VII: The Theory of Normal Cycles 1.10 Part VIII: Applications to Curves and Surfaces 1 4 6 Part I Motivations Motivation: Curves 2.1 The Length of a Curve 2.1.1 The Length of a Segment and a Polygon 2.1.2 The General Definition 2.1.3 The Length of a C1 -Curve 2.1.4 An Obvious Convergence Result 2.1.5 Warning! Negative Results 2.2 The Curvature of a Curve 2.2.1 The Pointwise Curvature of a Curve 2.2.2 The Global (or Total) Curvature 2.3 The Gauss Map of a Curve 2.4 Curves in E2 2.4.1 A Pointwise Convergence Result for Plane Curves 2.4.2 Warning! A Negative Result on the Approximation by Conics 2.4.3 The Signed Curvature of a Smooth Plane Curve 13 13 13 14 15 16 16 17 17 19 21 22 22 22 24 v vi Contents 2.5 2.4.4 The Signed Curvature of a Plane Polygon 26 2.4.5 Signed Curvature and Topology 27 Conclusion 28 Motivation: Surfaces 3.1 The Area of a Surface 3.1.1 The Area of a Piecewise Linear Surface 3.1.2 The Area of a Smooth Surface 3.1.3 Warning! The Lantern of Schwarz 3.2 The Pointwise Gauss Curvature 3.2.1 Background on the Curvatures of Surfaces 3.2.2 Gauss Curvature and Geodesic Triangles 3.2.3 The Angular Defect of a Vertex of a Polyhedron 3.2.4 Warning! A Negative Result 3.2.5 Warning! The Pointwise Gauss Curvature of a Closed Surface 3.2.6 Warning! A Negative Result Concerning the Approximation by Quadrics 3.3 The Gauss Map of a Surface 3.3.1 The Gauss Map of a Smooth Surface 3.3.2 The Gauss Map of a Polyhedron 3.4 The Global Gauss Curvature 3.5 The Volume 29 29 29 29 30 33 33 34 36 37 39 40 41 41 42 43 44 Part II Background: Metrics and Measures Distance and Projection 4.1 The Distance Function 4.2 The Projection Map 4.3 The Reach of a Subset 4.4 The Voronoi Diagrams 4.5 The Medial Axis of a Subset 47 47 49 52 55 55 Elements of Measure Theory 5.1 Outer Measures and Measures 5.1.1 Outer Measures 5.1.2 Measures 5.1.3 Outer Measures vs Measures 5.1.4 Signed Measures 5.1.5 Borel Measures 5.2 Measurable Functions and Their Integrals 5.2.1 Measurable Functions 5.2.2 Integral of Measurable Functions 57 57 57 58 58 59 60 60 60 61 Contents 5.3 5.4 5.5 5.6 5.7 vii The Standard Lebesgue Measure on EN 5.3.1 Lebesgue Outer Measure on R and EN 5.3.2 Lebesgue Measure on R and EN 5.3.3 Change of Variable Hausdorff Measures Area and Coarea Formula Radon Measures Convergence of Measures 62 63 64 64 65 66 67 67 Part III Background: Polyhedra and Convex Subsets Polyhedra 6.1 Definitions and Properties of Polyhedra 6.2 Euler Characteristic 6.3 Gauss Curvature of a Polyhedron 71 71 74 75 Convex Subsets 7.1 Convex Subsets 7.1.1 Definition and Basic Properties 7.1.2 The Support Function 7.1.3 The Volume of Convex Bodies 7.2 Differential Properties of the Boundary 7.3 The Volume of the Boundary of a Convex Body 7.4 The Transversal Integral and the Hadwiger Theorem 7.4.1 Notion of Valuation 7.4.2 Transversal Integral 7.4.3 The Hadwiger Theorem 77 77 77 79 80 81 82 84 84 85 86 Part IV Background: Classical Tools in Differential Geometry Differential Forms and Densities on EN 8.1 Differential Forms and Their Integrals 8.1.1 Differential Forms on EN 8.1.2 Integration of N-Differential Forms on EN 8.2 Densities 8.2.1 Notion of Density on EN 8.2.2 Integration of Densities on EN and the Associated Measure 91 91 91 93 94 94 95 Measures on Manifolds 9.1 Integration of Differential Forms 9.2 Density and Measure on a Manifold 9.3 The Fubini Theorem on a Fiber Bundle 97 97 98 99 viii Contents 10 Background on Riemannian Geometry 101 10.1 Riemannian Metric and Levi-Civita Connexion 101 10.2 Properties of the Curvature Tensor 102 10.3 Connexion Forms and Curvature Forms 103 10.4 The Volume Form 103 10.5 The Gauss–Bonnet Theorem 104 10.6 Spheres and Balls 104 10.7 The Grassmann Manifolds 105 10.7.1 The Grassmann Manifold Go (N, k) 105 10.7.2 The Grassmann Manifold G(N, k) 106 10.7.3 The Grassmann Manifolds AG(N, k) and AGo (N, k) 107 11 Riemannian Submanifolds 109 11.1 Some Generalities on (Smooth) Submanifolds 109 11.2 The Volume of a Submanifold 112 11.3 Hypersurfaces in EN 113 11.3.1 The Second Fundamental Form of a Hypersurface 113 11.3.2 kth -Mean Curvature of a Hypersurface 114 11.4 Submanifolds in EN of Any Codimension 115 11.4.1 The Second Fundamental Form of a Submanifold 115 11.4.2 kth -Mean Curvatures in Large Codimension 116 11.4.3 The Normal Connexion 116 11.4.4 The Gauss–Codazzi–Ricci Equations 117 11.5 The Gauss Map of a Submanifold 118 11.5.1 The Gauss Map of a Hypersurface 118 11.5.2 The Gauss Map of a Submanifold of Any Codimension 118 12 Currents 121 12.1 Basic Definitions and Properties on Currents 121 12.2 Rectifiable Currents 122 12.3 Three Theorems 124 Part V On Volume 13 Approximation of the Volume 129 13.1 The General Framework 129 13.2 A General Evaluation Theorem for the Volume 131 13.2.1 Statement of the Main Result 131 13.2.2 Proof of Theorem 38 131 13.3 An Approximation Result 133 13.4 A Convergence Theorem for the Volume 135 13.4.1 The Framework 135 13.4.2 Statement of the Theorem 137 Contents ix 14 Approximation of the Length of Curves 139 14.1 A General Approximation Result 139 14.2 An Approximation by a Polygonal Line 140 15 Approximation of the Area of Surfaces 143 15.1 A General Approximation of the Area 143 15.2 Triangulations 144 15.2.1 Geometric Invariant Associated to a Triangle 144 15.2.2 Geometric Invariant Associated to a Triangulation 145 15.3 Relative Height of a Triangulation Inscribed in a Surface 145 15.4 A Bound on the Deviation Angle 146 15.4.1 Statement of the Result and Its Consequences 146 15.4.2 Proof of Theorem 45 147 15.5 Approximation of the Area of a Smooth Surface by the Area of a Triangulation 150 Part VI The Steiner Formula 16 The Steiner Formula for Convex Subsets 153 16.1 The Steiner Formula for Convex Bodies (1840) 153 16.2 Examples: Segments, Discs, and Balls 155 16.3 Convex Bodies in EN Whose Boundary is a Polyhedron 158 16.4 Convex Bodies with Smooth Boundary 159 16.5 Evaluation of the Quermassintegrale by Means of Transversal Integrals 161 16.6 Continuity of the Φk 162 16.7 An Additivity Formula 164 17 Tubes Formula 165 17.1 The Lipschitz–Killing Curvatures 165 17.2 The Tubes Formula of Weyl (1939) 168 17.2.1 The Volume of a Tube 168 17.2.2 Intrinsic Character of the Mk 170 17.3 The Euler Characteristic 171 17.4 Partial Continuity of the Φk 171 17.5 Transversal Integrals 172 17.6 On the Differentiability of the Immersions 174 18 Subsets of Positive Reach 177 18.1 Subsets of Positive Reach (Federer, 1958) 177 18.2 The Steiner Formula 180 18.3 Curvature Measures 182 18.4 The Euler Characteristic 182 18.5 The Problem of Continuity of the Φk 184 18.6 The Transversal Integrals 186 x Contents Part VII The Theory of Normal Cycles 19 Invariant Forms 189 19.1 Invariant Forms on EN × EN 189 19.2 Invariant Differential Forms on EN × SN−1 190 19.3 Examples in Low Dimensions 192 20 The Normal Cycle 193 20.1 The Notion of a Normal Cycle 193 20.1.1 Normal Cycle of a Smooth Submanifold 194 20.1.2 Normal Cycle of a Subset of Positive Reach 194 20.1.3 Normal Cycle of a Polyhedron 195 20.1.4 Normal Cycle of a Subanalytic Set 196 20.2 Existence and Uniqueness of the Normal Cycle 196 20.3 A Convergence Theorem 198 20.3.1 Boundness of the Mass of Normal Cycles 199 20.3.2 Convergence of the Normal Cycles 199 20.4 Approximation of Normal Cycles 200 21 Curvature Measures of Geometric Sets 205 21.1 Definition of Curvatures 205 21.1.1 The Case of Smooth Submanifolds 206 21.1.2 The Case of Polyhedra 208 21.2 Continuity of the Mk 209 21.3 Curvature Measures of Geometric Sets 210 21.4 Convergence and Approximation Theorems 210 22 Second Fundamental Measure 213 22.1 A Vector-Valued Invariant Form 213 22.2 Second Fundamental Measure Associated to a Geometric Set 214 22.3 The Case of a Smooth Hypersurface 215 22.4 The Case of a Polyhedron 216 22.5 Convergence and Approximation 216 22.6 An Example of Application 217 Part VIII Applications to Curves and Surfaces 23 Curvature Measures in E2 221 23.1 Invariant Forms of E2 × S1 221 23.2 Bounded Domains in E2 221 23.2.1 The Normal Cycle of a Bounded Domain 221 23.2.2 The Mass of the Normal Cycle of a Domain in E2 223 23.3 Plane Curves 224 23.3.1 The Normal Cycle of an (Embedded) Curve in E2 224 23.3.2 The Mass of the Normal Cycle of a Curve in E2 225 Contents xi 23.4 The Length of Plane Curves 226 23.4.1 Smooth Curves 226 23.4.2 Polygon Lines 227 23.5 The Curvature of Plane Curves 227 23.5.1 Smooth Curves 227 23.5.2 Polygon Lines 228 24 Curvature Measures in E3 231 24.1 Invariant Forms of E3 × S2 231 24.2 Space Curves and Polygons 231 24.2.1 The Normal Cycle of Space Curves 231 24.2.2 The Length of Space Curves 232 24.2.3 The Curvature of Space Curves 233 24.3 Surfaces and Bounded Domains in E3 234 24.3.1 The Normal Cycle of a Bounded Domain 234 24.3.2 The Mass of the Normal Cycle of a Domain in E3 235 24.3.3 The Curvature Measures of a Domain 236 24.4 Second Fundamental Measure for Surfaces 238 25 Approximation of the Curvature of Curves 241 25.1 Curves in E2 241 25.2 Curves in E3 242 26 Approximation of the Curvatures of Surfaces 249 26.1 The General Approximation Result 249 26.2 Approximation by a Triangulation 250 26.2.1 A Bound on the Mass of the Normal Cycle 250 26.2.2 Approximation of the Curvatures 251 26.2.3 Triangulations Closely Inscribed in a Surface 252 27 On Restricted Delaunay Triangulations 253 27.1 Delaunay Triangulation 253 27.1.1 Main Definitions 253 27.1.2 The Empty Ball Property 254 27.1.3 Delaunay Triangulation Restricted to a Subset 255 27.2 Approximation Using a Delaunay Triangulation 256 27.2.1 The Notion of ε -Sample 256 27.2.2 A Bound on the Hausdorff Distance 256 27.2.3 Convergence of the Normals 257 27.2.4 Convergence of Length and Area 258 27.2.5 Convergence of Curvatures 258 Bibliography 261 Index 265 26.2 Approximation by a Triangulation 251 Theorem 78 The mass of the normal cycle of C is bounded as follows: ) ≤ 2αB lB |B×E αB v (C) v e M(N ) ≤ 4π sin lB n n |B×E M(Ne (C) M(N(C) |B×E ) ≤ A(B) + 2αB lB + 4π sin2 αB lB nv ne M(∂ (N(C) |B×E )) ≤ l∂ B + 2αB nv∂ Proof of Theorem 78 Over an edge e, the support of the normal cycle is reduced to a portion of cylinder, which can be identified with the product of e by an arc of circle c of the 2-sphere S2 The angle spanned by any point of c with the normal ξ of the surface at any vertex of e is smaller than αB The result follows by summing over all the edges of B Over a vertex v, the support of the normal cycle lies in v × S2 We bound its mass as follows: consider two adjacent faces belonging to the 1-ring of v The normals of these two faces span with ξv a geodesic triangle in S2 whose area is smaller than 4π sin2 αB Moreover, the mass of the normal cycle over v is smaller than the sum of the areas of these geodesic triangles Since the 1-ring of v contains a number of edges less than or equal to ne , the mass of the normal cycle over v is smaller than ne αB Now, since B contains nv vertices, M(Nv (C) |B×E ) ≤ 4π sin2 αB v e lB n n Since the normal cycle over a face can be identified with the face itself, one gets part (3) by summing the terms over the faces, the edges, and the vertices The boundary of N(C) is composed of edges corresponding to the edges of |B×E ∂ B, and arcs of circles above the vertices belonging to ∂ B We deduce part (4) 26.2.2 Approximation of the Curvatures We can now approximate the curvature measures of S with the curvature measures of T , as an immediate consequence of Theorem 67 Theorem 79 Let K be a compact subset of E3 whose boundary is a smooth (closed oriented embedded) hypersurface S Let C be a compact subset of E3 whose boundary is a triangulation T closely near S Let B be the interior of a union of triangles of T Then, |MC (B) − MK (pr(B))| ≤ H H max(δB , αB )( supB (1, |hB |) αB e ) (A(B) + 2αB (lB + nvδ ) + 4π sin2 n + l∂ B ); − δB |hB | 252 26 Approximation of the Curvatures of Surfaces |MC (B) − MK (pr(B))| ≤ G G max(δB , αB )( supB (1, |hB |) αB e ) (A(B) + 2αB (lB + nvδ ) + 4π sin2 n + l∂ B ); − δB |hB | |hC (B) − hK (pr(B))| ≤ max(δB , αB )( supB (1, |hB |) αB e ) (A(B) + 2αB (lB + nvδ ) + 4π sin2 n + l∂ B ) − δB |hB | 26.2.3 Triangulations Closely Inscribed in a Surface Let us now make a stronger assumption: the triangulation is closely inscribed in the surface S We have seen in Chap 15 that, under this assumption, one can control the angular deviation in terms of the shape of the triangles of T Let us give a simpler version of Theorem 79, introducing the circumradius r(t) of each triangle t of the triangulation Theorem 80 Under the assumption of Theorem 79: • |MC (B) − MK (pr(B))| ≤ CS Kε , H H • |MC (B) − MK (pr(B))| ≤ CS Kε , G G • |hC (B) − K (pr(B))| ≤ CS Kε , where: • CS is a real number depending only on the maximum curvature of S • K = ∑t∈T,t⊂B r(t)2 + ∑t∈T,t⊂B,t∩∂ B=0 r(t) / • ε = max{r(t),t ∈ T,t ⊂ B} We will see in Chap 27 that, in several cases, the number K can be bounded from above, implying that the curvature measures of a sequence of increasingly fine triangulations of a smooth surface converge to those of the smooth surface Remark These theorems can be interpreted as follows: suppose that one deals with a triangulated mesh T This mesh can be considered as the approximation of an infinity of smooth surfaces S Although it is in general impossible to evaluate the geometry of S without other assumptions, Theorems 79 and 80 claim that every smooth surface S, in which T is closely inscribed and • whose normal vector field is close to the normal of the faces, • whose second fundamental form is “not too big,” has a local geometry close to that of T Moreover, the error between the mutual curvatures is bounded by an explicit constant depending on the intrinsic geometry of T and on the two previous assumptions Chapter 27 On Restricted Delaunay Triangulations We deal here with particular approximations of smooth curves or surfaces in EN (N = or N = 3): those which arise from Voronoi diagrams and Delaunay triangulations of EN Although everything can be done in any dimension, we restrict ourself to E2 and E3 for simplicity We refer to [16, 18, 19, 38] for a large and deep study of Voronoi diagrams, Delaunay triangulations, and more generally for surface reconstruction 27.1 Delaunay Triangulation 27.1.1 Main Definitions A sample S denotes simply a subset of points of EN (which can be finite or infinite) In this section, we assume that the points of S are finite and in general position, i.e.: • If N = 2, no subset of four points of S lies on a same circle of E2 • If N = 3, no subset of five points of S lies on a same sphere of E3 We defined the notion of triangulation in Chap We improve it a little by defining a triangulation associated to a finite set of points Definition 55 Let S be a finite set of points of EN A triangulation of S is a simplicial cell complex embedded in EN whose set of vertices is S, and such that the union of its cells is the convex hull conv(S) of S Using Sect 4.1, we can define the Voronoi diagram Vor(S) associated to any sample S of EN , as a cell decomposition of EN into convex tetrahedra (or triangles if N = 2) The Delaunay triangulation associated to the Voronoi diagram of S is a particular triangulation of S, dual to Vor(S) Recall the classical definition of duality between cell complexes: two cell complexes V and D are dual if there exists an involutive correspondence between the 253 254 27 On Restricted Delaunay Triangulations faces of V and the faces of D that reverses the inclusions, i.e., for any two faces f and g of V , their dual faces f ∗ and g∗ of D satisfy f ⊂ g =⇒ g∗ ⊂ f ∗ Suppose for instance that N = If f is a face (of any dimension k(0 ≤ k ≤ 3)) of the Voronoi diagram of a sample S, all points of the interior of f have the same closest points in S Let S f ⊂ S denote the subset of those closest points The face dual to f is the convex hull of S f Its dimension is − k Definition 56 The Delaunay triangulation Del(S) of S is the simplicial complex consisting of all the faces dual to Vor(S) If we restrict our attention to E3 , we can simply claim that the Delaunay triangulation associated to S is the simplicial complex whose vertices are the points of S and which decomposes the convex hull of S as follows: the convex hull of four points of S defines a three-dimensional cell if the intersections of the corresponding Voronoi cells are nonempty As before, one defines the Delaunay tetrahedra, Delaunay faces (or triangles), and Delaunay edges The Delaunay vertices are nothing but the points of S If S is in general position, then Del(S) is a triangulation of S in the sense of Definition 55 The reader may adapt these considerations if the sample S lies in E2 27.1.2 The Empty Ball Property We still assume that S lies in E3 As an obvious consequence of the definition, the (relative) interior of a Voronoi k-face f is the set of points having exactly − k + nearest points of the sample S Consequently, there exists a ball empty of points of S, whose boundary is a sphere containing the vertices of the simplex f ∗ dual to f One says that the simplex f ∗ has the empty ball property Any Delaunay tetrahedron corresponds a unique empty ball However, there is a continuous family of empty balls corresponding to a Delaunay face (triangle) or Delaunay edge The following proposition characterizes the Delaunay triangulations in terms of the empty ball property Proposition 23 A triangulation T of a finite set of points S is a Delaunay triangulation of S if any 3-simplex of T has a circumscribing 2-sphere that does not enclose any point of S Any 3-simplex with vertices in S which can be circumscribed by a 2-sphere that does not enclose any point of S is a face of a Delaunay triangulation of S As before, the reader may adapt these considerations if the sample S lies in E2 27.1 Delaunay Triangulation 255 27.1.3 Delaunay Triangulation Restricted to a Subset Let S be a (finite) sample of E3 and let X be any subset of E3 Definition 57 If f is any k-face of Vor(S), f ∩ X is called the restriction of f to X The subcomplex Vor|X (S) of all nonempty restrictions of faces of Vor(S) to X is called the restriction of Vor(S) to X The restriction to X of the Delaunay triangulation Del(S) is the subcomplex DelX (S) of Del(S), which is the union of the k-faces of Del(S) (0 ≤ k ≤ 2) whose dual Voronoi (3 − k)-faces intersect X The main problem is to know if this new triangulation DelX (S) is a “good approximation” of X (in a sense to be specified) This is the objective of Sect 27.2 It will essentially depend on the position of S with respect to X Let us restrict our attention to the case of surfaces (Figs 27.1 and 27.2) Fig 27.1 Delaunay triangulation of a point set sampling a smooth curve Edges of the restricted triangulation are shown as solid blue lines, other Delaunay edges of the Voronoi complex are shown as red lines This image is courtesy of Steve Oudot, I.N.R.I.A Geometrica Fig 27.2 Restriction (in blue) of a Delaunay triangulation to a smooth algebraic surface (in green) Voronoi edges are shown in red This image is courtesy of Steve Oudot, I.N.R.I.A Geometrica 256 27 On Restricted Delaunay Triangulations 27.2 Approximation Using a Delaunay Triangulation As we have seen in Chap 26, the approximation of the geometric invariants of a surface by another close surface is “good” if the corresponding normals are “close.” If a smooth surface S is approximated by a Delaunay triangulation restricted to it, as constructed in the previous sections, we must ensure that the normals of the faces of the triangles are close to the normals of the surface S (at the corresponding points) 27.2.1 The Notion of ε -Sample In a series of papers, Amenta et al [5, 7] gave sufficient conditions for a sample of points on a surface to be interesting, in terms of the local feature size lfs defined in Chap Let us summarize them Definition 58 Let ε be a real number such that < ε < A set of points S on X is an ε -sample of X if and only if, for every point m of X, the ball B(m, ε lfs(m)) encloses at least one point of S.1 An ε -sample of X will generally be denoted by Sε 27.2.2 A Bound on the Hausdorff Distance 27.2.2.1 The Case of a Smooth Curve of EN Let c be a smooth (connected) curve of EN and let S be a sample of c We say that S is in natural position with respect to c if it satisfies the following (natural) conditions: • S contains at least three points and two of them are the end points of c • No vertex for Vor (S) lies on S If an ε -sample Sε of a curve c is in natural position, then Delc (Sε ) tends to with ε More precisely, we have the following result Theorem 81 Let c be a smooth (connected) curve of EN and let Sε be an ε -sample of c in natural position Then, Delc (Sε ) is a polygonal curve homeomorphic to c, and the Hausdorff distance between c and Delc (Sε ) satisfies d(Delc (Sε ), c) ≤ 2ε sup lfs(m) m∈c The proof of this theorem, its improvements, and related results can be found in [6, 16, 39] The reader must be careful: some authors replace B(m, ε lfs(m)) by B(m, ε ) in Definition 58, which implies confusion in many theorems We will only use Definition 58 in this chapter 27.2 Approximation Using a Delaunay Triangulation 257 27.2.2.2 The Case of a Smooth Surface The previous result can be extended to surfaces in E3 Boissonnat and Oudot [17] proved the following theorem Theorem 82 Let S be a smooth surface in E3 and Sε be an ε -sample of S Then, if ε < 0.18: DelS (Sε ) is closely inscribed in S (and consequently, S and Sε are homeomorphic) Moreover, the Hausdorff distance between DelS (Sε ) and S satisfies2 d(DelS (Sε ), S) ≤ 4.5ε sup lfs(m) m∈S 27.2.3 Convergence of the Normals The following result shows that if ε is smooth enough, then the restricted Delaunay triangulation associated to the sample S is “in good position.” 27.2.3.1 The Case of a Smooth Curve For curves, the following classical result (see for instance [38, Lemma 2.4, p 29]) shows that the angular deviation between a smooth curve c and Delc (Sε ) tends to with ε Theorem 83 Let c be a smooth (connected) curve of E2 and let Sε be an ε -sample of c in natural position Then: The angular deviation αmax of Delc (Sε ) with respect to c satisfies sin αmax ≤ ε 1−ε In particular, αmax = O(ε ) 27.2.3.2 The Case of a Smooth Surface Theorem 83 can be extended to surfaces [5, 38] There is a slight difference between Theorem 82 and Theorem 4.5 of [17], in which ε must be smaller than 0.091 This is due to the fact that the authors introduce and deal with the concept of loose ε -sample, weaker than the usual one 258 27 On Restricted Delaunay Triangulations Theorem 84 Let Sε be an ε -sample on S Then, the angular deviation αmax of DelS (Sε ) with respect to S satisfies αmax = O(ε ) 27.2.4 Convergence of Length and Area By definition, the length of a smooth curve is the supremum of the lengths of the polygonal lines closely inscribed in it We have seen many times that this result has no immediate generalization to surfaces (see the Lantern of Schwarz in Chap 3) However, if we restrict our attention to ε -samples, we have easy convergence theorems: as an immediate consequence of Theorem 84, we get the following result Theorem 85 Let S be a closed surface of E3 Let DelS (Sε ) be the restricted Delaunay triangulation of an ε -sample Sε with respect to S Then lim A(DelS (Sε )) = A(S) ε →0 27.2.5 Convergence of Curvatures Unfortunately, we cannot get further results on the convergence of the curvature measures of DelS (Sε ) to those of S without additional assumptions In fact, to apply Theorem 66 for instance, we need to bound the mass of the normal cycle DelS (Sε ) In general, this mass is not bounded when ε tends to (see [35] for a counterexample) That is why we need a stronger assumption on the sample As an example, one can use sequences of κ -light samples Definition 59 An ε -sample Sε on a surface S is κ -light if, for every point m ∈ S, the ball B(m, ε lfs(m)) encloses at most κ -points of S (Fig 27.3) Fig 27.3 An ε -sample S of a closed curve in E , where ε = : for each point m of the curve, the circle of center m and radius lfs(m) contains at least one point of S m’ m 27.2 Approximation Using a Delaunay Triangulation 259 Corollary 18 Let DelS (Sε ) be the restricted Delaunay 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circumradius, 144 closely inscribed, 136 closely near, 129 coarea formula, 66 connexion form, 103 contact structure, 112 convergence of the normal cycles, 199 convex body, 77 convex hypersurface, 77 convex subset, 77 countably additive, 59 covariant derivative, 113 current, 121 curvature (of a curve), 17 curvature form, 103 curvature measure, 182, 205, 210 curvature tensor, 102 Delaunay triangulation, 253 density, 94 density on a manifold, 98 deviation angle, 130, 146, 200 differential form, 91 dihedral angle, 73 distance function, 47 edge, 71 empty ball property, 254 Euler characteristic, 74 existence of the normal cycle, 196 external dihedral angle, 74 face, 71 fatness, 135, 144 Fenchel-Milnor theorem, 20 flat norm, 123 Frnet equations, 28 Frnet frame, 28 Fubini theorem, 99 fundamental (N − 1)-form, 214 Gauss curvature, 33, 75 Gauss map, 118 Gauss–Bonnet formula, 104 Gauss-Bonnet theorem, 75 geodesic, 34 geodesic triangle, 34 geometric subset, 197 Grassmann manifold, 105 Hadwiger theorem, 84 Hausdorff distance, 48 Hausdorff measure, 65 homotopy, 124 hypersurface, 113 265 266 inscribed, 136 integral (current), 123 integral of a measurable function, 61 internal dihedral angle, 73 invariant form, 189 Kubota formula, 87 Lagrangian submanifold, 112 Lantern of Schwarz, 30 Lebesgue measure, 64 Lebesgue outer measure, 63 Lebesgue theorem, 62 Legendrian, 112 length (of a curve), 13 Levi-Civita connexion, 101 lines of curvature, 33 Liouville form, 111 Lipschitz submanifold, 122 Lipschitz–Killing curvature, 159, 165 local feature size, 55 mass (of a current), 123 mass of the normal cycle, 199 measurable function, 60 measurable space, 58 measure, 57 measure on a manifold, 98 medial axis, 55 normal bundle, 109 normal cone, 73 normal connexion, 116 normal curvature, 117 normal cycle, 193 normalized external dihedral angle, 74 normalized internal dihedral angle, 73 orthogonal projection, 49 outer measure, 57 polyhedron, 71 principal directions, 33 projection map, 49 Quermassintegrale, 154 Index Radon measure, 67 reach, 52, 177 relative curvature, 133 relative height, 145 restricted Delaunay triangulation, 255 Ricci tensor, 102 Riemannian geometry, 101 Riemannian metric, 101 rightness, 144 sample, 253 scalar curvature, 102 second fundamental form, 33 second fundamental measure, 214 sectional curvature, 102 signed curvature (of a curve), 24 signed measure, 59 simple curve, 20 simple measurable function, 60 simplicial complex, 71 sphere, 104 Steiner formula, 153 submanifold, 109 subset with positive reach, 177 support (of a current), 121 support function, 80 symplectic form, 111 torsion (of a connexion), 101 transversal integral, 85 triangle, 71 triangulation, 144, 253 tube, 48, 165 tubes formula, 168 tubular neighborhood, 48 tubular neighborhood theorem, 110 uniqueness of the normal cycle, 196 valuation, 85 vector valued invariant form, 213 vertex, 71 volume (of a submanifold), 112 Voronoi diagram, 55, 253 Voronoi region, 55 weakly regular, 200 Weingarten tensor, 33 ... examples and counterexamples to the problem of convergence of geometric quantities: the length of a smooth curve and its curvature in Chap and the area of a smooth Introduction surface and its mean and. .. bodies, and the Hadwiger theorem [52] 1.6 Part IV: Background – Classical Tools on Differential Geometry Chapters and recall the definition of differential forms and densities on a manifold, and their... Chapter Motivation: Curves The length and the curvature of a smooth space curve, the area of a smooth surface and its Gauss and mean curvatures, and the volume and the intrinsic (resp., extrinsic)